A Fixed Point Approach to Certain Convex Programs with Applications in Stochastic Programming

This paper characterizes a certain convex program, which often arises in connection with stochastic programming as a fixed point problem, and explores the possibility of solving it by applying simplicial algorithms for finding fixed points of point-to-set mappings. We first establish equivalence between the convex program and the fixed point problem for some point-to-set mapping. As this mapping may have unfavorable properties from a computational viewpoint, we then modify it to reconstruct a point-to-set mapping and consider the application of a simplicial fixed point algorithm to the latter mapping. It is shown that the algorithm converges to a fixed point of the mapping and yields an optimal solution to the original convex program under certain conditions.